Velocity Limit in DPD Simulations of Wall-Bounded Flows

Transcription

1 Velocit Limit in DPD Simulations of Wall-Bounded Flows Dmitr A. Fedosov, Igor V. Pivkin and George Em Karniadakis Division of Applied Mathematics, Brown Universit, Providence, RI 2912 USA Abstract Dissipative particle dnamics (DPD) is a relativel new mesoscopic simulation approach, which has been successfull applied in modeling comple fluids in periodic domains. A recent modification [1] has allowed DPD simulations of polmers for realistic values of the Schmidt number. However, DPD and its etensions encounter difficulties in simulating even simple fluids in wall-bounded domains. The two main problems are wall boundar conditions and compressibilit effects the topic of the present work which limit the application of DPD to low Renolds number (Re) flows (e.g., Re < 1). Here, we establish an empirical criterion that provides an upper limit in velocit and correspondingl in Re for a fied computational domain, assuming a deviation from Navier-Stokes solutions of at most 5%. This limit could be increased b increasing the size of the computational domain at approimatel linear computational cost. Results are presented for the driven-cavit flow reaching, for first time, Re = 1, and similar etensions can be established for other comple-geometr flows. A sstematic investigation is presented with respect to both different tpes of boundar conditions and compressibilit effects for the standard DPD method as well as the modified version that simulates highl viscous fluids. Ke words: no-slip, compressibilit, boundar conditions, DPD 1 Introduction Dissipative particle dnamics (DPD) is a mesoscopic method [2,3] that can potentiall bridge the gap between atomistic and continuum descriptions in fluids. The DPD particles represent clusters of molecules moving together in a Corresponding author: gk@dam.brown.edu Preprint submitted to Elsevier Science 26 September 29

2 Lagrangian fashion subject to soft quadratic potentials. In contrast to molecular dnamics method, DPD emplos much larger time steps and particle sizes because of the soft particle interactions. In particular, the DPD method appears to be successful in simulations of comple fluids, such as suspensions of polmers, DNA, and colloids in a Newtonian incompressible solvent, etc., see [4 6]. Unlike other methods, such as Smoothed Particle Hdrodnamics (SPH) [7] and Lattice Boltzmann Method (LBM) [8], the DPD model includes thermal fluctuations, which ma pla a crucial role even in mesoscopic flows. Such corrections have to be eplicitl introduced in LBM [9] and SPH [1]. In addition to man simulations of comple fluids, several attempts have been made to simulate simple fluids at finite Renolds number (Re > 1 but tpicall less than 1) for several prototpe flows, e.g. flows past clinders, spheres, and inside channels and cavities [11 15]. The connection of DPD to molecular dnamics as well as to Navier-Stokes equations was investigated in [16]. It has been found that DPD ehibits hdrodnamic behavior and agrees well with numerical solutions of the Navier-Stokes equations but the question remains when this agreement breaks down and, moreover, what is the level of acceptable accurac. Unlike continuum-based discretizations where the simulation codes tpicall blow up for under-resolving a flow field (essentiall acting as a diagnostic measure), in the Lagrangian-based DPD simulations there is no blow-up but the simulation ma converge to an erroneous flow state - a far more dangerous situation! An eample of a DPD simulation is shown in figure 1 for the triangular-cavit flow at Renolds number Re = 1 along with spectral element simulations [17] emploing eighth-order polnomials (here, the Renolds number is based on the lid velocit and the height of the cavit.) The figure shows streamlines of the cavit flow with the upper wall moving to the right. In the absence of inertia, the triangular-cavit flow ehibits a cascade of counter-rotating vortices, with their strength decaing eponentiall as we approach the bottom corner of the domain (left plot). Hence, capturing these weak secondar vortices is challenging, and, indeed, most standard grid-based methods fail to resolve more than two or three vortices. The effect of inertia appears to modif the flow structure and onl two vortices are visible in the spectral element simulations at Re = 1 (middle plot). In this case, DPD also resolves two vortices although the secondar vorte appears to be under-resolved. In fact, sstematic comparisons between DPD and spectral element solutions showed good agreement of the flow fields at low Re but the results were strongl depended on the boundar conditions emploed in DPD to enforce the no-slip condition. At higher values of Re, the DPD results ield relativel large discrepanc compared to the spectral element solutions depending on the specific boundar conditions emploed and the velocit of the driven lid. The problems encountered in the DPD simulation of the triangular-cavit flow are tpical in most DPD simulations of wall-bounded flows, namel: (1) The 2

3 SEM, Stokes SEM, Navier-Stokes DPD Fig. 1. Streamline patterns from simulations of the triangular-cavit flow. The upper wall is moving to the right and the angle between the two stationar walls is 3. Left: Stokes flow simulated with spectral elements. Middle: Re = 1 flow simulated with spectral elements. Right: Re = 1 flow simulated with DPD. specific implementation of boundar conditions affects the maimum Renolds number that can be simulated for a specific domain, and (2) Densit fluctuations and associated clustering of particles arise, especiall close to walls, which ma also affect the bulk of the flow. Compressibilit effects are common in other methods (e.g., SPH or LBM) but also different boundar condition models have been shown to greatl affect the flow patterns. These effects have also been identified in [12] as limiting the accurac of DPD for flows past arras of spheres and clinders. In the current paper, we investigate sstematicall the aforementioned two issues and develop a simple criterion to diagnose erroneous DPD results in wall-bounded flows. To this end, we consider the square-cavit flow for which there is vast literature of solutions based on discretizations of Navier-Stokes equations. In addition to the standard DPD method, we present results obtained with a modified version of DPD, which was recentl introduced in [1]. In this version, the relative dissipative and random force interactions are modified (see section 2 for details) in order to increase the the viscosit of the DPD fluid and hence the Schmidt number, which is defined as Sc = ν, where ν is D the kinematic viscosit of the fluid and D is the diffusivit. As pointed out in [18], the standard DPD is able to reproduce fluids with values of Schmidt numbers of order O(1), in contrast to O(1 3 ) Schmidt numbers of real fluids. This increase in Schmidt number is desired in polmer flows and other transport problems. The paper is organized as follows. In the net section we review briefl the DPD governing equations and in section 3 we compare various no-slip boundar conditions. In section 4 we present simulation results for the cavit flow, and subsequentl we propose a quantitative criterion (based on the number 3

4 densit) that determines the maimum Renolds number that can be simulated accuratel with DPD. We conclude in section 5 with a brief discussion. 2 DPD Governing Equations The DPD sstem consists of N point particles of mass m i, position r i and velocit v i. These particles represent molecular clusters rather than individual atoms and interact through simple pairwise-additive forces. The total force eerted on a particle i b particle j consists of three terms: (1) conservative, (2) dissipative, and (3) random forces, i.e., F C ij = F C ij (r ij )ˆr ij, (1) F D ij = γω D (r ij )(v ij ˆr ij )ˆr ij, (2) F R ij = σω R (r ij )ξ ijˆr ij, (3) where r ij = r i r j, r ij = r ij, ˆr ij = r ij /r ij, and v ij = v i v j. The coefficients γ and σ determine the amplitude of dissipative and random forces, respectivel. ω D and ω R are weight functions, ξ ij is a normall distributed random variable with zero mean, unit variance, and ξ ij = ξ ji. All forces act within a sphere of radius r c, the cutoff radius, which defines the length scale in the DPD sstem. The conservative force Fij C (r ij ) is tpicall given b F C ij (r ij ) = a ij (1 r ij /r c ) for r ij r c, for r ij > r c, where a ij = a i a j and a i, a j are conservative force coefficients for particles i and j, respectivel. The DPD sstem relaes to an equilibrium temperature T when the random and dissipative forces satisf the fluctuation dissipation theorem [19] (4) ω D (r ij ) = [ ω R (r ij ) ] 2 (5) σ 2 = 2γk B T, (6) where k B is the Boltzmann constant. The tpical choice for the weight functions is (1 r ω R ij /r c ) p for r ij r c, (r ij ) = (7) for r ij > r c. where p = 1 for the standard DPD method. However, other choices for these envelopes have been proposed in order to increase the Schmidt number of 4

5 the DPD fluid, e.g., p =.25 in [1]. We will refer to this version of DPD as modified DPD in the current paper. The time evolution of velocities and positions of particles is described b Newton s second law of motion dr i = v i dt, (8) dv i = 1 (F C m ijdt + F D ijdt + Fij R dt). (9) i j i There are several methods that can be used for the integration of DPD evolution equations. In this work we use two different time-integration algorithms in order to compare their performance in simulations of wall-bounded sstems. The selected methods are (1) modified velocit-verlet algorithm [3], and (2) self-consistent DPD integration scheme [2]. We want to investigate if an of these time-integration algorithms ields results with a different upper limit in Re. 3 Boundar Conditions One of the main issues in DPD simulation of wall-bounded flows is the correct imposition of boundar conditions. There are two main approaches for modeling solid boundaries: indirect methods, which avoid direct modeling of the phsical boundaries b reling on modifications of the periodic boundar conditions or of the domain [21,22]; and direct methods, which model solid boundaries b locall freezing regions of particles [23,24]. The first categor is limited to simple flows, so here we focus on wall boundar conditions in DPD, which emplo collections of frozen particles in combination with particle reflection rules. In particular, reflection of fluid particles at the fluid-solid interface prevents fluid particles from penetrating the wall; this can be implemented b using specular, bounce-back, bounce-forward, or Mawellian reflection. Here we emplo several different (no-slip) wall boundar conditions. Revenga et al. [24] performed simulations of DPD flows without an conservative interactions (a ij = ) and showed that large values of the densit of wall particles enforces the no-slip condition at the fluid-solid interface. In [25] the no-slip boundar condition was achieved b emploing a bounce-back reflection even for relativel small wall densities. However, in the presence of repulsive interactions (a ij ) non-negligible densit fluctuations are observed in the near-wall region. More recentl, it has been shown [15] that these fluctuations induce an apparent slip due to the shearing between high and low densit laers and it becomes more pronounced at higher shear rates. Hence, one has 5

6 to simultaneousl impose the no-slip condition directl at the solid boundar and, in addition, to control the near-wall densit fluctuations. Specificall, a force boundar condition (FBC) was developed in [15] b freezing several wall laers of uniforml distributed (simple cubic lattice) DPD particles in combination with bounce-back reflection at the interface. In addition, the repulsive interactions from the wall particles are adjusted according to the fluid and wall number densities, as follows: a w = 1 [.39(n f k B T +.1a f n 2 ] 2 f), (1) a f.33n 2 w n w.8536 where a w and a f are the conservative force coefficients of wall and fluid particles, and n w, n f are wall and fluid number densities, respectivel. Such tuning of the conservative force does not completel solve the problem of near-wall densit fluctuations, however, it eliminates slip for low and moderate shear rates in the standard DPD method ( corresponding to p = 1 in equation (7)). Results from our various DPD simulations reveal that the problem of densit fluctuations close to the wall remains in the case of freezing laers of particles. This artifact is due to an imbalance of the DPD forces between wall particles and surrounding fluid particle along the wall-normal direction. (The dissipative and random forces have no net contributions along the wall-normal direction unlike the conservative force as will be shown at the end of this section.) Therefore, the idea of introducing an adaptive wall potential seems to be reasonable, so following [26] we use here two tpes of adaptive boundar conditions, which we will call ABC- and ABC-I. In ABC-, we freeze several wall laers of DPD particles similar to FBC, but the conservative force of wall particles is set to zero. In order to compensate the repulsive force imbalance from the wall we introduce adaptivel a wall force acting on the fluid particles. The adaptive force calculation, which is described in detail in [26], controls the densit fluctuations and can enforce an desirable densit profile close to the wall. In ABC-I, densit fluctuations are controlled similar to ABC-. However, instead of laers of frozen DPD particles, we introduce image particles in the wall region as it was done in [13,26]. For completeness, we describe below the procedure for computing the random and dissipative force contributions for ABC-I. Let us consider a wall perpendicular to -ais and located at = W ; the wall is moving with velocit v W remaining on the = W plane. We assume that there are N W fluid particles within the cutoff distance r c from the wall. The i-th particle has coordinates ( i, i, z i ) and velocit v i while the total force acting on particle i is F i. It is convenient to introduce a ghost particle g, although it is not necessar to construct it eplicitl in the code. The dissipative and random force contributions of the wall boundar conditions at each time step are epressed using the following pseudo-code: 6

7 for particle i = 1,.., N W create ghost particle g with g = i + ξ, g = 2 W i, z g = z i + ξ z and v g = 2v W v i. for particle j = i + 1,.., N W compute F R jg, F D jg F j = F j + F D jg + dt 1/2 F R jg end end F i = F i F D jg dt 1/2 F R jg The random variables ξ and ξ z are uniforml distributed in the interval [ r c, r c ]. In addition, when fluid particles penetrate into the wall region, we perform a bounce-back reflection of these particles into the fluid region. In the current work, we target uniform densit profiles, i.e., the near wall densit is equal to the bulk densit. We have found that for a uniform densit profile close to the wall the iterative adaptive procedure for the conservative force in ABC converges to the wall force that corresponds to a potential profile, which compensates for the imbalance of repulsive forces from the surrounding semi-spherical fluid region. We have verified that the same potential can be obtained b numerical integration of the conservative force multiplied b the radial distribution function over the semi-spherical fluid region an approach first presented in [27] i.e., F w (h) = n Fw C (r)g(r)dv, (11) V s\v cap(h) where h is the distance from the fluid particle to the wall, n is the number densit, V s is the sphere volume, V cap (h) is a spherical cap (cut out) from the wall, F C w (r) is the conservative DPD force in the direction normal to the wall, and g(r) is the radial distribution function. The epression for F w (h) cannot be obtained analticall because the radial distribution function does not, in general, have an analtical epression. Hence, for a uniform densit profile the wall adaptive force can be imposed during the simulation b either using an iterative procedure or b numerical integration after computing the radial distribution function for a specific DPD fluid. An alternative wa to impose the no-slip boundar condition (based on frozen wall particles) is to consider in a pre-processing stage a flow simulation at the equilibrium state (no net fluid motion) in combination with an adaptive wall 7

8 h Vav FLUID SUBREGION WALL L WALL SUBREGION L Vav Fig. 2. Sketch illustrating the concept of equilibrium boundar condition with shear (EBC-S). shear procedure. At the pre-processing stage, the domain which covers both fluid and solid wall regions is assumed to be periodic in all directions and without an solid walls present. The DPD particles are distributed in a lattice or randoml in the domain and simulations are run until the equilibrium state is reached. The particles, which occup the region of solid walls, are then frozen at some instant of time and later used to model solid walls in combination with bounce-back reflection at the fluid-solid interface. We have developed a procedure with two tpes of boundar conditions, which we denote as EBC- when the adaptive wall shear procedure is not used and EBC-S when the adaptive procedure is emploed. The details of EBC-S tpe can be eplained using the sketch of figure 2. In simulations, we consider subregions of the computational domain of width L = r c adjacent to the fluid-solid interface in both fluid and wall regions. We divide the fluid and wall subregions into bins of height h, whose value can be chosen based on the desired accurac of the near-wall velocit profile. These subregions can be divided into bins along the wall if the velocit profile changes in the direction parallel to the wall. During the simulation, in each bin in the fluid subregion the time-averaged velocit v av is collected over a specified number of time-steps. The velocities of the particles inside each bin in the wall subregion are set to be opposite, i.e. v av, to the average velocit in the corresponding fluid subregion bin, which is smmetric with respect to the fluid-solid interface. The wall particles (shown as open circles) do not move in the simulations and carr onl velocit information, which is used in the calculation of the dissipative force. We find that this model enforces the noslip boundar conditions and eliminates densit fluctuations in the near-wall region, similarl to ABC-I. Moreover, the EBC-S model has about the same computational cost as the two previous models (FBC and ABC) and it can 8

9 be easil implemented. FBC Force boundar condition, equation (1) ABC- ABC-I EBC- Adaptive boundar condition with frozen particles Adaptive boundar condition with image particles Equilibrium boundar condition without shear correction EBC-S Equilibrium boundar condition with shear correction Table 1 Summar of no-slip wall boundar conditions. In summar, all three tpes of boundar conditions (i.e., FBC, ABC and EBC, see table 1) are effective for flows involving low to moderate wall shear rates. However, in the high shear rate regime onl ABC-I and EBC-S enforce properl the no-slip condition at the walls with minimum near-wall densit fluctuations. 3.1 Modified DPD: More Viscous Fluids In the discussion so far we have considered the standard DPD method, which simulates fluids with Schmidt number of order one [3]. In a recent paper [1], an attempt was made to achieve much larger values of the Schmidt number and one of the modifications proposed was a change of the weight function of dissipative and random forces. In particular, a slower deca with distance of the interaction envelope enhances the dissipative interactions of particles with surrounding particles and substantiall increases the viscosit of the fluid and hence the corresponding Schmidt number. In addition, in order to increase the value of viscosit we can increase the cutoff radius in the DPD simulation. Here, we are interested to eamine how accurate are the three aforementioned tpes of boundar conditions for this modified DPD method. To this end, we performed several DPD simulations of Poiseuille flow in a channel. In the case of the standard DPD method all tpes of boundar conditions give correct results for modest levels of wall shear stress. In the modified DPD simulations we emploed the following parameters: number densit n = 4; force coefficients a = 18.75, γ = 4.5, σ = 3.; temperature k B T = 1.; cutoff radius r c = 1.; domain size 2 2 1; pressure drop g =.5; and eponent in the random force weight function p =.25 (equation (7)). For comparison, the viscosit in the modified DPD simulations is approimatel twice the viscosit of the standard DPD case while the Schmidt number is about si times larger. Figure 3 summarizes the results from several modified DPD simulations. The solution corresponding to ABC-I and EBC-S coincides with the analtical solution. However, in the absence of the adaptive shear procedure, 9

10 4.5 v FBC ABC- ABC-I EBC- EBC-S analtic solution number densit FBC ABC- ABC-I EBC- EBC-S analtic solution Fig. 3. Modified DPD (p =.25): Comparison of different boundar conditions. Left: Velocit profiles. Right: Densit profiles. Onl ABC-I and EBC-S give correct results. i.e., case EBC-, we obtain a solution with a small slip at the wall. Also, for simulations emploing FBC and ABC- we observe an under-development of the Poiseuille profile. To understand these results we consider each DPD force contribution from the wall separatel for the three directional components: normal, streamwise tangential and across-stream tangential. First, we compute them for an ideal implementation of the wall using the periodic Poiseuille flow method (PPFM) [22]. In this method the domain is subdivided into two subdomains and two counter-flowing Poiseuille flows are established b appling a force with the same magnitude but opposite direction in each subdomain. The planes separating the subdomains can be considered as imaginar ideal walls, i.e., walls that are modeled without eplicit implementation of the no-slip conditions. At each time step we measure the instantaneous force acting on the particles within distance r from these ideal walls on one side from the particles on the opposite side. We find that onl two components of force contributions out of nine have a non-zero average value, i.e., the normal component of the conservative force and the streamwise (tangential) component of the dissipative force. The are shown as a function of distance from the wall with solid lines in figure 4. The normal component of the conservative force acting on the fluid particles from the wall controls the fluid densit fluctuations close to the wall. The streamwise dissipative force component determines the streamwise velocit profile close to the wall. In the same figure, we plot the average values of the forces acting on fluid particles from the wall for different implementations of the wall boundar conditions considered above. We observe that the average values of the normal component of the conservative force are close to the ideal profile for all methods ecept for FBC. Specificall, FBC leads to results with large densit fluctuations compared to other methods (see fig- 1

11 ure 3). The average streamwise dissipative force components of the ABC-I and EBC-S capture the ideal profile quite well. For other methods we observe deviations, which are consistent with the simulation results for the velocit profiles shown in figure FBC ABC- ABC-I EBC- EBC-S PPFM (ideal) wall 2 FBC ABC- ABC-I EBC- EBC-S PPFM (ideal) wall C n F 15 1 D s F r r Fig. 4. Average wall forces acting on the fluid particles for different implementations of wall boundar conditions as a function of distance from the wall. Left: Normal component of the conservative force. Right: Streamwise component of the dissipative force. 4 Simulation Results In this section we present results from DPD simulations of the square lid-driven cavit flow at different Renolds numbers; the Renolds number is defined as Re = UL, where U is the velocit of the moving lid, L is the height of the ν cavit, and ν is a kinematic viscosit of the fluid. The DPD simulation results will be compared to numerical solutions obtained b the highl accurate spectral element discretization of the Navier-Stokes equations [17]. All DPD simulations were performed using the modified version of the velocit-verlet algorithm [3] for the integration of equations of motion. For comparison purposes, we also emploed the self-consistent DPD algorithm [2]; the difference in the results was negligible, i.e., of the order of the statistical fluctuations. 4.1 Simulation parameters The following parameters are used in DPD simulations. The fluid number densit is n = 3 and the cutoff radius r c = 1. The conservative force coefficient of fluid particles is a = 25. while the random and dissipative force coefficients 11

12 are σ = 3. and γ = 4.5. The temperature is set to k B T = 1 satisfing the fluctuation-dissipation relations (5) and (6). The kinematic viscosit ν of the DPD fluid is equal to.2854; it was obtained b fitting a parabola to the DPD results from periodic Poiseuille flow method (PPFM) described in [22]. The majorit of simulations emploed the modified velocit-verlet integration scheme with λ =.5 [3], which corresponds to the standard velocit-verlet scheme widel used in Molecular Dnamics simulations. The time step is between.1 and.1 depending on the Renolds number of a particular case. With regards to boundar conditions, for FBC and ABC- the solid walls are modeled b freezing several laers of uniforml distributed DPD particles. The number densit of the walls is n w = 8.6 while the conservative force coefficient of wall particles is a w =.117 according to equation (1) [15]. For EBC- and EBC-S the solid walls are modeled b freezing DPD particles at equilibrium while the densit of the walls is the same as that of fluid. For ABC- and ABC-I we use a bin grid for all walls in order to compute the densit fluctuations. In particular, the bins have the length of the cavit wall and height rc =.2. We are targeting uniform densit profile with densit 5 magnitude n = 3. In each corner we have two overlapping grids, which result in non-uniform densit profile in the corner as we will observe later. For EBC- S we use a bin grid for all walls in order to compute the near-wall velocit profile and mimic a counter flow in the wall region. In addition, we emploed a bounce-back reflection at the fluid-solid interface. For comparison, we have run several cases with specular and Mawellian reflection; the results showed no dependence on the particular reflection tpe. The flow domain size is (in DPD units), and it is periodic in the z direction. Another set of DPD simulations was performed on a larger domain, corresponding to (in DPD units). The large domain size in the z direction allowed us to obtain converged statistics relativel quickl after the stead state was reached. The domain is subdivided into 4 bins in and directions. The simulations are run for 5, time steps and statistical data are collected starting from time step 5,. The highest Renolds number in the current stud, Re = 1, was obtained on a domain Reference numerical solution In figure 5 we show streamlines for the square-cavit flow at Renolds number Re = 1 and Re = 1. The solution was obtained using the solver NEKTAR [17] based on spectral element discretization of the incompressible Navier-Stokes equations using 9 quadrilateral elements with fourth- and sith-order polnomials, for the low and high Renolds numbers, respectivel. The velocit of the upper wall is fied while no-slip conditions are applied at all the other walls. 12

13 VV DD VV DD 1 v 1 v Fig. 5. Streamline pattern obtained using spectral element discretizations at (left) Re = 1 and (right) Re = 1. The upper wall is moving to the right. Comparison of results is performed along the cuts VV and DD. The coordinates are normalized b the domain size. 4.3 Low Renolds number flow We start our DPD tests from the case of static fluid corresponding to Re = according to our definition. 1 We can use these equilibrium simulations in order to identif differences among the various solid-wall boundar condition models. In figure 6 we plot the number densit profiles etracted along the vertical (VV ) and diagonal (DD ) lines defined in figure 5. The results of simulations with FBC and ABC- are shown. The left plot in the figure shows that the use of FBC contributes to large densit fluctuations in the near-wall region. However, densit fluctuations can be smoothed out if we use the ABC- tpe. On the right plot we see that we have even larger densit fluctuations in the corners due to the presence of two intersecting walls. For ABC- and ABC- I we have a significant improvement of the number densit in the corner. For EBC- and EBC-S tpes we find that the densit profile is essentiall uniform. The static fluid simulations reveal the effect of appling different boundar conditions. In the following, we will identif one more effect when the fluid is in motion, namel compressibilit. Therefore, it is important to consider in DPD simulations of wall-bounded flows the boundar condition effect on densit fluctuations separatel, and note that certain implementations ma eliminate this effect while for other implementations strong boundar effects ma be present. 1 We note that Re = here does not correspond to Stokes flow, which is the tpical notation. 13

14 1 Re =, VV cut 1 Re =, DD cut 8 FBC ABC- bulk densit 8 FBC ABC- bulk densit number densit 6 4 number densit Fig. 6. Static fluid. Densit profiles etracted along VV and DD lines. The -coordinate is normalized b the domain size. We first performed DPD simulations in the regime of relativel small Renolds numbers, i.e., 1 < Re < 5, using different tpes of boundar conditions. In figure 7 we present simulation results for the flow at Re = 25. Results obtained 1 Re = 25, VV cut.4 Re = 25, DD cut.8 SEM DPD, FBC.3.2 SEM DPD, FBC velocit v v velocit v v Fig. 7. Velocit profiles etracted along the VV and DD lines. Re = 25 b the spectral element method (SEM) are plotted with lines, and DPD results are shown with smbols. We have a ver good agreement between the DPD and the continuum-based simulations, which is tpical in this range Re 5. The discrepanc in the corner for the DD cut can be eplained b the lack of DPD resolution there, where a ver sharp velocit gradient is present. We also note that DPD can be used to accuratel simulate flows in the range of low Renolds numbers, i.e., Re < 1. However, computationall it can become ver epensive to obtain a smooth solution because it requires a long-time averaging. 14

15 4.4 Moderate Renolds number flow Net we increase Renolds number further but within the range 5 < Re < 4, and find that the DPD results using FBC give a noticeable discrepanc compared to the spectral element results. In order to investigate the reasons for the DPD failure in this case we consider the fluid densit fluctuations in the corner where the moving wall encounters the stationar vertical wall. Figure 8 shows densit profiles etracted along the DD line. This plot shows 15 number densit 1 5 Re = 1 Re = 25 Re = 5 Re = 1 Re = 2 Re = 4 dense region growth n= Fig. 8. Densit profiles etracted along DD line for different Re. Force boundar condition (FBC). The coordinates are normalized b the domain size. that at the corner = = 1 the maimum densit increases with Re for the fied computational domain (L is fied) or equivalentl with the velocit of the moving wall. This is due to compressibilit effects allowing for a particle accumulation in the corner as the particles moving along the upper wall encounter an obstacle represented b the stationar vertical wall. We also note that approimatel up to Re = 5 the densit profile around the corner region remains similar to that of a static fluid, even though the maimum number densit at the corner increases. For higher Renolds numbers (Re > 5) the densit profile in the corner starts changing drasticall, and the local dense region grows in size and leads to modification of the global flow structure, and hence it affects the large vorte at the center of the cavit. Net, we perform a series of DPD simulations using ABC- to model the wall boundaries. The new results show that the densit fluctuations at the corner = = 1 are substantiall reduced, hence the agreement of DPD results with the continuum-based solutions is valid in a wider range of Renolds numbers, 15

16 e.g., < Re < 1. Figure 9 shows densit profiles etracted along the DD line. As we also found in Poiseuille flow, ABC- improves the near-wall densit 15 number densit 1 5 Re = 1 Re = 25 Re = 5 Re = 1 Re = 2 Re = 4 Re = 5, FBC dense region growth n= Fig. 9. Densit profiles etracted along DD line for different Re. Adaptive boundar condition (ABC-). The coordinates are normalized b the domain size. fluctuations, and therefore densit fluctuations are less pronounced compared to simulations emploing FBC. Moreover, since we emplo here the standard DPD method other boundar condition models (e.g., ABC-I, EBC-, or EBC- S) have similar performance to ABC Effect of domain size In order to simulate higher Re number flows we can increase the size of the computational domain. To this end, we performed several simulations for the cavit domain of size These results indicate that we can reach an accurate solution for Renolds number twice as big compared to the 1 1 domain. A further analsis of this will be given below. In figure 1 we present tpical simulation results for Re = 1 using FBC to model the walls for the 1 1 and 2 2 domains. We have alread demonstrated that the domain 1 1 with FBC can be used to accuratel simulate flows in the range < Re < 5. However, for the domain 2 2 with FBC we are able to use DPD for a somewhat wider range, i.e., < Re < 1, at comparable accurac. 16

17 1 Re = 1, VV cut.4 Re = 1, DD cut.8.6 SEM DPD, 11 DPD, 22 v.2 SEM DPD, 11 DPD, 22 velocit.4.2 v velocit -.2 v v Fig. 1. Effect of domain size. Velocit profiles etracted along the VV and DD lines. The coordinates are normalized b the domain size. (Re = 1) 4.6 Modified DPD Here we stud the performance of the modified DPD (p =.25 in equation (7)) [1] and compare it with the standard DPD (p = 1 in equation (7)). The kinematic viscosit in this case increases to.547. All other DPD simulation parameters remain the same as before, and we onl have to adjust the velocit of the moving wall in order to match the targeted value of the Renolds number. Figure 11 compares our simulation results for the cavit flow at Re = 5 using EBC-S for different eponents p = 1. (standard DPD) and p =.25 (modified DPD). We note that the results for p =.25 are not ver 1 Re = 5, VV cut.4 Re = 5, DD cut.8.6 SEM DPD, p=.25 DPD, p=1. v SEM DPD, p=.25 DPD, p=1. velocit.4.2 v velocit v v Fig. 11. Velocit profiles etracted along the VV and DD lines for standard (p = 1) and modified (p =.25) DPD methods. The coordinates are normalized b the domain size. (Re = 5) 17

18 accurate compared to the results for p = 1.. This is due to the increase of viscosit of a fluid which requires us to set a higher velocit for the moving wall in order to achieve Re = 5. The compressibilit effects depend onl on the repulsive interactions and not on the strength of the thermostat, i.e. the eact form of the dissipative and random DPD forces. A conclusion which can be drawn from these tests is that it is advantageous to use low viscosit in the DPD simulations instead of high velocit to achieve a certain Re value. With regards to boundar conditions, we emploed EBC-S tpe here based on the aforementioned results for channel flow, i.e., EBC-S performs well both for the standard and modified DPD version. In order to quantif the effect of different tpes of boundar conditions for the modified DPD method, we also run simulations using ABC-. Figure 12 shows corresponding results at Re = 25. The use of ABC- leads to larger errors in the solution as this 1 Re = 25, VV cut.4 Re = 25, DD cut.8.6 SEM DPD, ABC DPD, EBC-S v SEM DPD, ABC DPD, EBC-S velocit.4.2 v velocit v v Fig. 12. Modified DPD: Velocit profiles etracted along the VV and DD lines for ABC- and EBC-S. The coordinates are normalized b the domain size. (Re = 25) tpe of boundar condition adjusts properl the repulsive forces but not the dissipative forces. We also note that ABC-I performs similarl to EBC-S but FBC and EBC- also lead to large errors, essentiall confirming our initial conclusions based on the Poiseuille flow stud in section Number densit effect Groot & Warren have shown that a sufficientl high number densit (n > 2) is required in DPD simulations in order to obtain a good approimation for the fluid obeing a quadratic equation of state [3]. The have also derived a formula which relates the dimensionless compressibilit k 1 of the fluid to the 18

19 number densit, conservative force coefficient, and temperature as follows k 1 = 1 + 2αan k B T (α =.11). (12) In principle, the number densit chosen for the simulation is a free parameter, however, often the value n = 3 is used as a lowest possible densit due to increasing computational cost for larger densities. Further, we investigate the issue of compressibilit and boundar condition effects for the cavit flow utilizing different fluid number densities. We have performed a series of simulations for the cavit flow in the regime < Re < 2 using bulk number densities n = 3, 6 and 9 for the small domain size 1 1, and also bulk number densities n = 3 and 6 for the larger domain size 2 2. In all simulations we fi the dimensionless compressibilit and temperature and adjust the conservative force coefficient to the bulk number densit using equation (12). Specificall, we run simulations for three sets of those parameters: (1) n = 3, a = 25, (2) n = 6, a = 12.5, and (3) n = 9, a = We note that we maintain the same fluid dimensionless compressibilit in order to be able to definitevel separate the compressibilit effects from the boundar condition effects. The results presented so far suggest that the limit of DPD applicabilit can be determined b the maimum allowed velocit of the moving wall or alternativel b the maimum allowed number densit in the corner. However, it appears that this limit can be well characterized b the densit ratio n ma /n where n ma is the maimum densit in the corner (averaged over the corner bin) and n is the bulk densit. Figure 13 shows the dependence of the densit ratio on the Renolds number for simulations in the small and large domains; each smbol in the plot corresponds to a separate cavit flow simulation. An important finding is that the number densit ratio varies approimatel linearl with Re. Furthermore, the lines which correspond to different number densities are parallel, hence the densit ratio has the same functional dependence for all bulk number densities, i.e. these lines have equal slopes. This, in turn, implies that the response of number densit ratio to a change in Re (or equivalentl to the velocit of the moving wall) is identical and independent of the bulk number densit. This fact supports the idea that the limit of DPD applicabilit can be determined b a criterion which is independent of the specific value of the bulk number densit used. The Re = case corresponds to zero velocit of the upper wall, hence this is an equilibrium state. The deviation of the DPD solution from the epected accurate solution (n ma /n = 1) for Re = is due solel to the boundar condition implementation. In particular, the boundar conditions emploed for the simulation results shown in figure 13 are of the FBC tpe. We see that as the bulk number densit increases the effect of the boundar condition is less pronounced. Simulations with the other tpes of boundar conditions (i.e., ABC or EBC) give the epected solution 19

20 14 n = 3, 11 n = 6, 11 n = 9, 11 ratio limit (small) 12 n = 3, 22 n = 6, 22 ratio limit (large) 1 densit ratio Re number Fig. 13. Limit of DPD applicabilit: Plotted is the number densit ratio n ma /n as a function of the Renolds number Re. The solid lines are for the 1 1 domain while the dash lines are for the 2 2 domain. The Re = case corresponds to the equilibrium state, i.e., static fluid. The walls are modeled using the FBC formulation. at Re =, i.e., the intersect the vertical ais at n ma /n 1. At this point, we can separate the effects of boundar conditions from the compressibilit effects, given the aforementioned parallel form of the number densit lines. In particular, the contribution of the boundar conditions can be etracted from the Re = case while the linear increase with Re in the maimum number densit is entirel due to compressibilit effects. Our simulation results show that the limit of DPD applicabilit can be determined b the maimum allowed number densit ratio, which is independent of the bulk number densit but depends on the geometr and to some degree on the size of the bin selected for statistical averaging. For eample for the cavit size 1 1 with 4 4 bins, we found that the maimum allowed number densit ratio for the standard DPD method is approimatel n ma /n = 6, see horizontal solid line in the figure. (This value is obtained b sstematic comparison of the velocit field from the DPD simulations with the reference spectral element solutions based on the incompressible Navier-Stokes equations.) The intersection of the horizontal line (defined b n ma /n = 6) with the inclined lines determines the maimum Re that can be accuratel simulated. This is demonstrated in figures 14 and 15, where we plot velocit profiles etracted along a vertical (VV ) and diagonal (DD ) cuts for Re = 5 and Re = 1, respectivel, emploing different number densities in simulations. Although the results for the case Re = 5, n = 6 are in a good agreement with the spectral element solution, the case Re = 5, n = 3 shows a noticeable dis- 2

21 crepanc in the results. We find a similar behavior in the case of Re = 1 for densities 6 and 9. Clearl, for selected domain size, Re = 5 and 1 are close to the maimum Renolds number values that can be simulated at number densit n = 6 and n = 9, respectivel. We observe that higher number densit allows one to simulate higher Re number flows with the FBC model. This is due to the fact that the boundar condition effects on the near-wall densit fluctuations is weaker for higher number densit flows. 1 Re = 5, VV cut.4 Re = 5, DD cut.8.6 SEM DPD, n=3 DPD, n=6 v SEM DPD, n=3 DPD, n=6 velocit.4.2 v velocit v v Fig. 14. Velocit profiles etracted along VV and DD lines for number densit 3 and 6. (Re = 5; small domain) 1 Re = 1, VV cut.4 Re = 1, DD cut.8.6 SEM DPD, n=6 DPD, n=9 v.2 SEM DPD, n=6 DPD, n=9 velocit.4.2 v velocit -.2 v v Fig. 15. Velocit profiles etracted along VV and DD lines for number densit 6 and 9. (Re = 1; small domain) In figure 13 we have included results from simulations in a larger cavit domain, and we observe that the slope of the lines corresponding to the 2 2 cavit is half that of the lines corresponding to 1 1 cavit. This is due to the fact that for the same Re the velocit of the horizontal wall is halved in the larger domain, and we have alread seen the strong dependence of the 21

22 accurac of the DPD simulation on the wall s velocit magnitude. However, in order to determine the true limit in Re in the larger domain simulations, we have to consider the bin size emploed in the statistical averaging. If we keep the same bin size as in 1 1 cavit we would have 8 8 bins and the DPD limit of n ma /n = 6 is valid, independent of the bulk number densit used. However, if we emplo a 4 4 grid of bins then the size of the bin is twice larger than the bin size for the small domain, and, hence, the densit fluctuations are reduced due to the wider area averaging. Using similar comparisons between the spectral element solutions and the DPD results processed on the large domain but with the 4 4 bin, we find that the limit of DPD applicabilit seems to be even higher, e.g., n ma /n 8. So in brief, these results show that two different criteria emploed in the comparison of DPD with the spectral element results ma lead to somewhat different values of the threshold ratio n ma /n universal curve densit ratio velocit of the moving wall Fig. 16. Universal curve for the maimum number densit ratio versus the driving velocit. We can now use the findings we just discussed in order to collapse all the results plotted in figure 13 into a single master curve. In particular, if we repeat the previous simulations with the more accurate boundar conditions (ABC or EBC) there will be no dependence on the nominal value of the bulk densit so all parallel curves collapse to a single curve passing through n ma /n = 1 in the vertical ais (at Re = ). Moreover, the slope of the curves depends on the wall s driving velocit, hence if we replace Re in the horizontal ais with the velocit of the wall U we should epect different domain sizes to also collapse onto one curve. This is, indeed, the case and the universal curve is shown in figure 16. In order to obtain the maimum value of velocit for which we obtain correct flow fields we propose, based on our results, an upper limit of n ma /n in the range [5, 1] for all cases. To test this criterion we performed parallel simulations (on 512 Blue Gene processors) at 22

23 a much higher Re = 1 on a domain using EBC-S boundar conditions. Here, we keep all simulation parameters the same and set the velocit of the moving wall to in order to match Re = 1. (The case with velocit v = corresponds approimatel to the densit ratio 5 in the figure 16, and, therefore, the DPD simulation results are epected to be accurate. ) In figure 17 we present simulation results at Re = 1 including spectral element simulations of incompressible and compressible (at nominal Mach number.3) flows. To prove that DPD accuratel captures the flow field 1 Re = 1, VV cut.4 Re = 1, DD cut.8.6 SEM, Ma= DPD, EBC-S SEM, Ma=.3.2 v SEM, Ma= DPD, EBC-S SEM, Ma=.3 velocit.4.2 v velocit -.2 v v Fig. 17. Velocit profiles etracted along the VV and DD lines. Re = 1. structure in this relativel high Renolds number case we present comparisons of the velocit field in terms of v and v contour plots in figure 18. In general, we observe a ver good agreement of the DPD results with the spectral element solutions. In particular, the DPD contours seem to lie between the contours of the incompressible and compressible solutions. We have also performed a similar simulation using the modified DPD, see figure 19. The velocit of the moving wall in this case is set to 5.47, which corresponds to approimatel 9.5 value of the densit ratio on the master curve. Here we see a more pronounced difference with the incompressible spectral element results, as epected in this case, due to the fact that compressibilit effects become more important. We find that the modified DPD contours are closer to the compressible Mach =.3 solution, however there is a large discrepanc of the results in the corner. The Re = 1 simulations verif that if the upper limit of n ma /n is in the range [5, 1] the DPD results will be accurate. This limit can be used in selecting an appropriate domain size for the desired Re number flow simulation. 23

24 Re = 1, v Re = 1, v Fig. 18. Contours of v (left) and v (right). SEM incompressible results are denoted b solid lines, compressible solution (Mach =.3) b dotted lines and standard DPD results b dashed lines. Re = 1. Re = 1, v Re = 1, v Fig. 19. Contours of v (left) and v (right). SEM incompressible results are denoted b solid lines, compressible solution (Mach =.3) b dotted lines and modified DPD results b dashed lines. Re = 1. 5 Summar In this paper we have studied the two fundamental issues that limit the maimum Renolds number that can be achieved in DPD simulations of wallbounded flows, namel, boundar conditions and compressibilit effects. In order to enforce accuratel wall boundar conditions in DPD, we have to properl model the interaction forces between the wall and the fluid particles. Specificall, the wall-normal component of the conservative force controls the densit fluctuations while the wall-tangential component of the dissipative 24

25 force enforces the correct velocit slope in the near-wall region. (The average random force is zero and does not pla an important role in the correct imposition of boundar conditions.) The models ABC-I and EBC-S that we studied make the proper force adjustments for both components whereas FBC, ABC- and EBC- modif onl the wall-normal conservative force component. The DPD simulations we presented here for Poiseuille flow as well as flow in a lid-driven cavit demonstrate that indeed ABC-I and EBC-S ield superior results. Densit fluctuations around the bulk densit in the case of the liddriven cavit ma achieve ver large values at the corners formed between two intersecting walls. We have developed a simple criterion to diagnose erroneous DPD results based on the ratio of the maimum densit in the domain over the bulk densit; the upper limit of this ratio should be in the range [5, 1]. To achieve accurate DPD results at higher Renold number, we propose to increase the size of the computational domain. Our criterion can be used to predict the minimum domain size to achieve a specific Renolds number. We have demonstrated this b simulating accuratel flow in the cavit at Re = 1 using both the standard as well as a modified version of DPD, the latter associated with highl viscous fluids. Increasing the domain size requires more DPD particles, so here we provide an approimate analsis of the associated computational cost. Specificall, the cost of DPD simulation per time-step is proportional to the number of particle pairs, since onl pairwise interactions are involved in the computation. Hence, the cost of the simulation with the simplest implementation would be proportional to N 2, where N is the number of particles in the simulation. However, the forces in DPD act locall within the cutoff radius r c, and, hence, a more effective implementation is to consider onl neighboring pairs (N n of them). Correspondingl, the cost per time-step is cost C N N n, where the constant C represents the number of operations required for each pair. We have verified this scaling in equilibrium simulations (periodic bo domain) using different number of particles N, with results revealing that cost N 1.1 if more than 1 4 particles are used. If we now increase the cavit domain size in and dimensions b k times the corresponding number of particles is k 2 N while the number of bins is k 2 N b. The computational cost per time step is cost Ck 2 N N n, so the simulation is k 2 times more epensive. However, we also need to account for an increase in the total number of time-steps required for the larger domains, namel, (i) the time t s required to reach a statisticall stead state, and (ii) the time t a required to perform the data averaging after the stead state is reached. To this end, we can estimate t s from the diffusion limit, i.e., t s L/ N ν, while t a N b steps where N N steps is the number of steps required to reach the desired accurac, and N b is the number of bins used for statistical processing. Hence, to reach stead state in the larger domain requires an etra cost b a factor of k 3 but the statistical averaging cost is constant for comparable accurac with the small domain. 25